83.38.9 problem 9

Internal problem ID [19397]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (C) at page 133
Problem number : 9
Date solved : Monday, March 31, 2025 at 07:12:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (3 \sin \left (x \right )-\cot \left (x \right )\right ) y^{\prime }+2 y \sin \left (x \right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.380 (sec). Leaf size: 44

ode:=diff(diff(y(x),x),x)+(3*sin(x)-cot(x))*diff(y(x),x)+2*sin(x)^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\frac {3 \cos \left (x \right ) \operatorname {csgn}\left (\csc \left (x \right )\right )}{2}} \left (c_1 \sin \left (\frac {\cot \left (x \right )}{2 \sqrt {-\csc \left (x \right )^{2}}}\right )+c_2 \cos \left (\frac {\cot \left (x \right )}{2 \sqrt {-\csc \left (x \right )^{2}}}\right )\right ) \]

✓ Mathematica. Time used: 0.129 (sec). Leaf size: 20

ode=D[y[x],{x,2}]+(3*Sin[x]-Cot[x])*D[y[x],x]+2*y[x]*Sin[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{\cos (x)} \left (c_2 e^{\cos (x)}+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*sin(x) - 1/tan(x))*Derivative(y(x), x) + 2*y(x)*sin(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (2*y(x)*sin(x)**2 + Derivative(y(x), (x, 2)))*tan(x)/(3*sin(x)*tan(x) - 1) + Derivative(y(x), x) cannot be solved by the factorable group method